Mathematica Eterna
Open Access

Abstract

Extremum principle for vector valued minimizers and the weak solutions of elliptic systems

Guo Kaili and Gao Hongya

In this paper we consider the minimum principle for vector valued minimizers of some functionals F(u; Ω) = Z Ω f(x, Du(x))dx. The main assumption on the density f(x, z)is a kind of ”monotonicity” with respect to the N × n matrix z. We also consider the maximum and minimum principle for weak solutions u of some elliptic systems − Xn i=1 Di(a α i (x, u(x)) = 0, x ∈ Ω, α = 1, . . . , N, and the main assumption on a α i (x, z) is 0 < Xn j=1 X N α=1 a α i (x, z)(z α i − z˜ α i ), where ˜z is a N × n matrix with respect to z.